Question 1 [2] Let G be a group. If H = {g 2 |g ∈ G} is a subgroup of G, prove that it is a normal subgroup of G.

Question 2 [5] Let m ∈ Z. If x m ∈ H, for every x ∈ G, then the order of every element in G/H is a divisor of m. Conversely, if the order of every element in G/H is a divisor of m, then x m ∈ H, for every x ∈ G. (Hint: If m and n are integers and n is positive, there exist unique integers q and r such that m = nq + r and 0 ≤ r < n.)

Question 3 [2,5] Use the Fundamental Homomorphism Theorem to prove that groups Z3 and Z6/h3i are isomorphic. (You need not prove that Z3 and Z6/h3i are groups.) Question 4 [3,5] Let H and K be normal subgroups of a group G, with H ⊆ K. Define φ : G/H → G/K by: φ(Ha) = Ka. Prove the following:

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